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IEEE/CAA Journal of Automatica Sinica

Pure State Feedback Switching Control Based on the Online Estimated State for Stochastic Open Quantum Systems

Shuang Cong, Zhixiang Dong

Published Sep 4, 2024• 2 citations
Publisher

Abstract

For the$n$-qubit stochastic open quantum systems, based on the Lyapunov stability theorem and LaSalle's invariant set principle, a pure state switching control based on on-line estimated state feedback (short for OQST-SFC) is proposed to realize the state transition the pure state of the target state including eigenstate and superposition state. The proposed switching control consists of a constant control and a control law designed based on the Lyapunov method, in which the Lyapunov function is the state distance of the system. The constant control is used to drive the system state from an initial state to the convergence domain only containing the target state, and a Lyapunov-based control is used to make the state enter the convergence domain and then continue to converge to the target state. At the same time, the continuous weak measurement of quantum system and the quantum state tomography method based on the on-line alternating direction multiplier (QST-OADM) are used to obtain the system information and estimate the quantum state which is used as the input of the quantum system controller. Then, the pure state feedback switching control method based on the on-line estimated state feedback is realized in an$n$-qubit stochastic open quantum system. The complete derivation process of$n$-qubit QST-OADM algorithm is given; Through strict theoretical proof and analysis, the convergence conditions to ensure any initial state of the quantum system to converge the target pure state are given. The proposed control method is applied to a 2-qubit stochastic open quantum system for numerical simulation experiments. Four possible different position cases between the initial estimated state and that of the controlled system are studied and discussed, and the performances of the state transition under the corresponding cases are analyzed.

DOI
10.1109/jas.2023.124071

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Abstract An online estimated state feedback control for trajectory tracking in stochastic open quantum systems is proposed in this paper, which is based on the Lyapunov‐based control method. By inducing the error between the controlled state, and the target state as the error state, the trajectory tracking problem of the quantum system is transformed into the error state transition control problem. The quantum state online estimation method QST‐OADM is applied to estimate the state of the error state system online, and the tracking control laws are designed by using the quantum Lyapunov stability theorem for driving the stochastic open quantum system from an arbitrary initial state to an arbitrary trajectory. The numerical simulation experiments and results analyses are given.
In this paper, we propose a Lyapunov-based state feedback control for state transfer based on the on-line quantum state estimation (OQSE). The OQSE is designed based on continuous weak measurements and compressed sensing. The controlled system is described by quantum master equation for open quantum systems, and the continuous measurement operators are derived according to the dynamic equation of system. The feedback control law is designed based on the Lyapunov stability theorem, and a strict proof of proposed control laws are given. At each sampling time, the state is estimated on-line, which is used to design the control law. The simulation experimental results show the effectiveness of the proposed feedback control strategy.
In this paper the state of Markovian open quantum system is estimated on-line based on the continuous weak measurement and compression sensing theory, and a quantum Lyapunov feedback control law is designed to transfer the state from the initial state to the desired target state based on the state estimated on-line. Numerical simulation experiments are implemented under MATLAB environment. The experimental results show the effectiveness of the on-line estimated state-based feedback control method proposed
A quantum state feedback control method is proposed in this paper. The state of a two-level open quantum system is estimated online based on the continuous weak measurement and the compressed sensing theory. Based on the state estimated online and the Lyapunov stability theorem, the state feedback control law used to transfer the quantum state is designed. Moreover, three numerical simulation experiments are implemented in the MATLAB environment: the state transfer from eigenstates to eigenstates, superposition states to superposition states, and superposition states to mixed states. The experimental results verify high performance of the proposed feedback control based on the state estimated online.
For stochastic quantum systems with measurement feedback, this article proposes a rapid switching control scheme based on state space partition and realizes the rapid stabilization of an eigenstate of an observable operator. Meanwhile, we apply the proposed scheme to the preparation of typical entangled states in multiqubit systems. In view of the convergence obstacle caused by the symmetric structure of the state space, especially in the case with degenerate observable operators, we first partition the state space into a subset containing the target state and its complement to distinguish the target state from its antipodal points, and then design the corresponding control laws in these two subsets, respectively, by using different Lyapunov functions. The interaction Hamiltonians are also constructed to drive the system state to the desired subset first, and further to the target state. In particular, the control law designed in the undesired subset guarantees the strictly monotonic descent of the corresponding Lyapunov function, which makes the system trajectory switch between the two subsets at most twice and has the potential to speed up the convergence process. We also prove the stability of the closed-loop system with the proposed switching control law based on the stochastic Lyapunov stability theory. By applying the proposed switching control scheme to a three-qubit system, we achieve the preparation of a GHZ state and a W state.
The global stabilization of an eigenstate for finite dimensional stochastic quantum systems with non-diagonal free Hamiltonian and non-regular measurement operator is studied in this paper. A two-part switching control strategy is proposed, in which a constant control is used to steer the system state to enter the convergence domain; another control law, which is designed based on Lyapunov stability theorem is used to attract the system state in convergence domain to the target state. The convergence of the target eigenstate from an arbitrary initial state is strictly proved. Moreover, the numerical experiments on a three dimensional stochastic quantum system are implemented to demonstrate the effectiveness of the control proposed.